TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET OF INDEPENDENT MEMORYLESS ON=OFF SOURCES in .NET

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TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET OF INDEPENDENT MEMORYLESS ON=OFF SOURCES
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11.1 11.1.1
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INTRODUCTION Long-Term Dependence and Packet Loss in Telecommunication Traf c
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In early literature about the performance of telecommunication systems, traf c was generally modeled as memoryless Poisson streams of packets. In these models the packet arrival processes show no time interdependence. Recent measurements on Web traf c show that this hypothesis is wrong and that Web traf c actually experiences what we now call long-term dependence. Long-term dependence is interesting not only because it contradicts Poisson's law, but also because it signi cantly impacts the performance of networks. One effect is that it dramatically increases packet loss in data networks. For example, let us focus on an Internet router. In a simple model, the router can be seen as a buffer served by a single server. When the buffer over ows, some packets are lost. The lost packets must be re-sent following TCP=IP, thus adding extra delay and traf c. If we simply model the router by a M =M =1 queue with an in nite buffer, input rate l, and service rate 1, then the probability pn that the queue length is greater than n is exactly ln. In a rst-order approximation, quantity pn can be identi ed with the packet loss rate in a buffer of size n. Therefore, to keep packet loss below some acceptable level e, it suf ces to make the buffer capacity greater than log e = log l),
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Self-Similar Network Traf c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.
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TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET
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that is, a logarithmic function of 1=e. In general, telecommunication designers work on e on the order of 10 6 and l < 0:8. Long-term dependent traf c can make loss and retry rates pn  Bn b for some b > 0 [1]. In other words, the queue size distribution has a heavy tail. Under this condition it is clear that buffer capacity would need to be raised to b=e 1=b to keep packet loss rate under the acceptable level e, which no longer leads to a logarithmic function of 1=e, but to a polynomial function of 1=e. Indeed, this minimal size would be several orders of magnitude higher than the capacity obtained with the Poisson model. In fact, actual router capacities are dangerously underestimated with regard to this new traf c condition. 11.1.2 Contribution of this
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The Poisson law is the natural consequence of the law of large numbers, best describes the cumulated effect of several independent, identically distributed (i.i.d) sources in parallel. Assume, for example, N sources, each of them producing on average l=N events per time unit according to a stationary random process. Then when N tends to in nity, the interevent times T tend to be i.i.d with a distribution function characterizing the Poisson law, PrfT > xg e lx : 11:1
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The convergence to a Poisson distribution still holds when the sources are not quite identical, as long as they have similar pro les. Feller [2] gave pretty general conditions for this convergence: basically, rst moments of the source interevent generation time must be O 1=N and second moments must be o 1=N . In this chapter we are interested in the case where traf c is created from a large set of independent sources that do not satisfy Feller's conditions. In particular, we focus on certain sets of on=off sources that produce long-term dependence when their sizes tend to in nity. It will also be shown that queues submitted to such sets of sources will experience buffer occupation with a polynomially decaying tail distribution. In the other chapters of this book it is assumed that some of the sources, taken individually, already produce long-term dependence. For example, some sources have heavy-tailed ``on'' periods. In this case the cumulated traf c shows long-term dependence and creates a heavy-tailed queue size distribution [3, 4]. The challenge in the present chapter is that none of the sources, taken separately, produces longterm dependence and a heavy-tailed queue size distribution, and that those phenomena eventually take place when the number of sources increases. To insist on this point, we will focus on individual on=off sources with memoryless pro les (exponentially distributed on periods and off periods). It has already been shown by Beran [5] and Jacquet [6] that such sources can create long-term dependence when their number increases. The contribution of the present chapter is to show that such sources can also create a polynomially decaying queue size distribution. We do not claim that pure memoryless on=off sources are necessarily realistic models for Web
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